An example of a relatively hyperbolic group pair is (G,{H}) where G is the fundamental group of a hyperbolic knot exterior, and H is the fundamental group of the deleted neighborhood of the knot. Another example is (A*B, {A,B}) where the star indicates free product. The first example admits no elementary splittings (splittings over virtually cyclic or parabolic subgroups), whereas the second does admit an elementary splitting.

Thurston's hyperbolic Dehn filling theorem has a group theoretic version which can be generalized to relatively hyperbolic group pairs, giving a way to produce new relatively hyperbolic pairs from old ones. We show that for all "sufficiently long" Dehn fillings of a relatively hyperbolic pair, the property of having no elementary splittings is preserved.

This is joint work with Daniel Groves.